[Paper Review] The $\partial\bar{\partial}$-lemma for general Clemens manifolds

Robert Friedman|arXiv (Cornell University)|Aug 2, 2017

Algebraic Geometry and Number Theory17 references3 citations

TL;DR

This paper establishes that the $\partial\bar{\partial}$-lemma holds for general small smoothings of three-dimensional Clemens manifolds—non-Kähler compact complex manifolds constructed by contracting rational curves with normal bundle $(-1,-1)$ in Calabi-Yau threefolds. Using mixed Hodge theory and variation of Hodge filtration in degenerations, the author proves the lemma holds on a nonempty open subset of the deformation space, confirming a key cohomological property for these manifolds.

ABSTRACT

We show that the $\partial\bar{\partial}$-lemma holds for the non-Kähler compact complex manifolds of dimension three with trivial canonical bundle constructed by Clemens as deformations of Calabi-Yau threefolds contracted along smooth rational curves with normal bundle of type $(-1, -1)$, at least on an open dense set in moduli. The proof uses the mixed Hodge structure on the singular fibers and an analysis of the variation of the Hodge filtration for the smooth fibers.

Motivation & Objective

• To establish the validity of the $\partial\bar{\partial}$-lemma for general small smoothings of Clemens manifolds, which are non-Kähler threefolds with trivial canonical bundle.
• To resolve the cohomological ambiguity in whether the Hodge-de Rham spectral sequence degenerates at $E_1$ and whether the resulting Hodge filtration is $k$-opposed.
• To analyze the variation of Hodge filtration and monodromy in degenerations of Calabi-Yau threefolds with ordinary double points.
• To determine whether the $\partial\bar{\partial}$-lemma holds in the context of singular fibers and their smoothings, particularly in the absence of Kähler structures.
• To clarify the cohomological structure of Clemens manifolds and their relationship to polarized Hodge structures of weight three.

Proposed method

• Utilizes mixed Hodge structures on singular fibers obtained by contracting rational curves with normal bundle $(-1,-1)$ in Calabi-Yau threefolds.
• Analyzes the variation of the Hodge filtration in the family of smoothings of the singular threefold $\overline{X}$, using monodromy data.
• Applies deformation theory for threefolds with ordinary double points and trivial dualizing sheaf to study the behavior of cohomology under smoothing.
• Employs induction on the number of contracted curves, reducing the problem to smaller configurations and using the fact that the $\partial\bar{\partial}$-lemma holds for smoothings of fewer curves.
• Relies on the degeneration of the Hodge-de Rham spectral sequence at $E_1$ and the assumption that the resolution of the singular fiber satisfies the $\partial\bar{\partial}$-lemma.
• Uses the fact that the $\partial\bar{\partial}$-lemma holds on a nonempty open subset of the deformation space, derived from the structure of the monodromy action and Hodge filtration variation.

Experimental results

Research questions

• RQ1 Does the $\partial\bar{\partial}$-lemma hold for general small smoothings of Clemens manifolds, which are non-Kähler threefolds with trivial canonical bundle?
• RQ2 Can the $\partial\bar{\partial}$-lemma be established for these manifolds despite the absence of a Kähler structure?
• RQ3 What is the role of mixed Hodge structures and variation of Hodge filtration in proving the $\partial\bar{\partial}$-lemma for singular fibers and their smoothings?
• RQ4 Is the $\partial\bar{\partial}$-lemma preserved under deformation equivalence, and can such manifolds be deformation equivalent to Kähler manifolds?
• RQ5 Does the existence of a balanced metric on Clemens manifolds imply the $\partial\bar{\partial}$-lemma, and if not, what stronger conditions are needed?

Key findings

• The $\partial\bar{\partial}$-lemma holds for general small smoothings of Clemens manifolds, specifically on a nonempty open subset of the deformation space.
• The proof relies on the variation of the Hodge filtration and monodromy action in degenerations of Calabi-Yau threefolds with ordinary double points.
• The result is established inductively by reducing the number of contracted curves and using the fact that the lemma holds for smoothings of fewer curves.
• The Hodge-de Rham spectral sequence degenerates at $E_1$ for the smooth fibers, and the resulting Hodge filtration is $k$-opposed, confirming the $\partial\bar{\partial}$-lemma.
• The $\partial\bar{\partial}$-lemma does not hold for all compact complex threefolds with trivial canonical bundle, as shown by counterexamples like certain Hopf surfaces.
• Clemens manifolds are not deformation equivalent to any compact complex manifold bimeromorphic to a Kähler manifold, due to their $b_2 = 0$ Betti number and the resulting contradiction in cohomological positivity.